ENTROPY GENERATION OF CONVECTION HEAT TRANSFER IN AN ASYMMETRICALLY HEATED PACKED DUCT

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ASYMMETRICALLY HEATED PACKED DUCT YASAR DEMIREL H.H. Ali B.A. Abu-Al-Saud Available at: https://works.

1 University of Nebraska - Lincoln From the SelectedWorks of YASAR DEMIREL 1997 ENTROPY GENERATION OF CONVECTION HEAT TRANSFER IN AN ASYMMETRICALLY HEATED PACKED DUCT YASAR DEMIREL H.H. Ali B.A. Abu-Al-Saud Available at:

2 Pergamon Int. Comm. Heat Mass Transfer, Vol. 24, No. 3, pp , 1997 Copyright 1997 Elsevier Science Ltd Printed m the USA. AU rights reserved ,/97 $ PII S (97) ENTROPY GENERATION OF CONVECTION HEAT TRANSFER IN AN ASYMMETRICALLY HEATED PACKED DUCT Y. Demirel, H. H. A1-Ali and B. A. Abu-AI-Saud Department of Chemical Engineering King Fahd University of Petroleum & Minerals, Dhahran Saudi Arabia (Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT An expression for the volumetric rate of entropy generation has been derived and displayed graphically to analyze the convection heat transfer for a fully established flow in a rectangular packed duct with LAt=I6 and H/W= The top and bottom walls of the duct are heated by constant, asymmetric heat fluxes, while the other walls are insulated that is the L2 thermal boundary conditions. Entropy generation maps reveal the regions where excessive entropy generation occurs due to physical and geometric parameters for a specified task within the system. Copyright 1997 Elsevier Science Ltd Introduction In an early study, CoIburn [ 1 ] reported that the rate of forced convection heat transfer from uniformly heated wall to air through a packed tube is about eight times higher than that of empty tube. Experimental investigation in a rectangular packed duct with unequal wall temperatures has also shown considerable increase in heat-transfer rate: Demirel [2] reported an increase of about twice in an air flow passage with L/H=8.3, and packed with Raschig ring, and Chrysler and Simons [3] found about ten times increase for liquid fluorocarbon-77 in a packed channel of spheres with L/H = 4.62, while, in a recent study, Hwang et al [4] reported an increase of three times for Freon-113 in a duct with L/H=9.0, H/W= and spherical packing Asymmetric thermal boundary conditions are common in thermal engineering and especially in electronic cooling [3], and all the possible thermal boundary conditions of convection heat transfer in a rectangular duct with constant heat flux are detailed by Gao and Hartnett [5]. In a recent study, Demirel and A1-AIi [6] derived an equation for the volumetric rate of entropy generation for fully developed flow in a packed channel with single surface heated by a constant 381

3 382 Y. Demirel, H.H. AI-Ali and B.A. Abu-Al-Saud Vol. 24, No. 3 heat flux (1L). This study concerns the effect of the asymmetric heating on the entropy generation in a packed air flow passage when two large surfaces (2L) are heated with constant, asymmetric heat fluxes. Entropy Generation The non equilibrium phenomenon of exchange of energy and momentum within the fluid and at the solid boundaries causes of continuous generation of entropy in the flow field. Local entropy generations per unit volume S'" of an incompressible Newtonian fluid for a twodimensional channel are represented by [7]: T2Lk.,~xJ +(~) +T 2(--~) showing the entropy generations due to finite temperature difference, and fluid friction by the first and second term respectively. Entropy generation profiles may be generated using equation (1) if the velocity and temperature fields are known in the heat transfer medium The duct under consideration is shown in Fig. 1,,,1/ Q~ y=h dy T ds"' T+dT dx Q2 y=o FIG. 1 Control volume of the rectangular duct It is assumed that the difference of wall-to-fluid bulk temperature is small enough, and there is no considerable change in physical properties of the fluid, no axial conduction, and no natural convection. Slug flow conditions (u = %) prevail through the cross section of the packed flow passage. The differential energy equation is

4 Vol. 24, No. 3 ENTROPY GENERATION IN A HEATED PACKED DUCT 383 (2> The heated walls are subjected to constant, asymmetric heat fluxes, which provide the necessary boundary conditions: at y = 0 at y = H -ke(c~t/oy) = Q2 = constant ke(cti'/oy) = Q1- constant The linearity of the energy equation (2) suggests that superposition methods may be employed to build solutions for asymmetric heating by adding the two fundamental solutions: (i) the surface one heated with the other insulated and (ii) the surface two heated with the other insulated. Hence the temperature profile for the system shown in Fig. 1 may be expressed by: where T= 1~(1+ r,4) (3) A = (hh/ 2k~)[(n + 1)Y 2-2nY 1] + (n + 1)(StX + 1) Y=y/H ; X=x/H ; ~=Q1/(hTo) ; n=q2/qt The effective thermal conductivity k e be expressed by: ke=k f e+(l-c) 0.2 3kp) P\I+--~ 2-) (4) which combines the static and dynamic contributions, shown by the first and second terms, respectively [8] The volumetric rate of entropy generation for the packed duct under consideration may be reduced from equation (1) and given by: Here the first term shows the entropy generated due to finite temperature difference S~r, while the entropy generated due to fluid friction S~ is shown by the second term. The ratio of these is: 0 = / (6) After inserting the velocity and temperature gradients which are detailed elsewhere [6] an expression for the volumetric rate of entropy generation may be derived and given by:

5 384 Y. Demirel, H.H. Al-Ali and B.A. Abu-AI-Saud Vol. 24, No. 3 (7) where K-00- ) I' e3pd2q ~ ~- o~sdpcpatb ; D = Dp/H In equation (7) local entropy generation has been expressed in terms of n, ~, Y, X and D including the properties of the fluid 9 and Cp The non dimensional volumetric rate of entropy generation J can be obtained as: key3 J= S'" Q21 =Jar+J~ (8) The dimensionless rate of entropy generation over the cross section, J' may be calculated by integration: 1 J'= [ Jdr (9) 0 In order to determine the total entropy generation in the duct, equation (8) must be integrated over the entire duct length. Results and Discussions In the calculations Prandtl number is assumed as 0.7 for the air flow. Y has changed between 0.02 and 0.98, while X varied between 3.0 and 16.0 as an attempt to lessen the entrance and the wall effects. A spherical packing of 4.8 cm in diameter with the void fraction of 0.58 is used. The top and bottom walls of the duct are heated by constant, unequal heat fluxes. For this thermal boundary condition of L2, the Nusselt number for a fully established flow between the parallel plates is given by Kays and Crawford [9]: 5385 Nu - (10) n Where n is the ratio of unequal constant heat fluxes satisfying the condition n<29 It was assumed that the average heat transfer coefficient is about three times higher than those given by the equation (9) after Hwang et al., [4] and Demirel [2] when the spherical packing is introduced into the air flow passage. With this assumption the temperature profiles for the packed duct have

6 Vol. 24, No. 3 ENTROPY GENERATION IN A HEATED PACKED DUCT 385 been estimated for Re=1867, St=0.013, Q1=43.9 W/m 2, To=300.4 K, n = 0.2 and 06, and shown in Fig ~ yo ~ 0.8~/~J/ " I,~,,. 5 1 ~ /,. 0 ~ 7 yo, 60.8~'~ FIG. 2 Temperature profiles for the rectangular packed duct (a) n=q2 " (b) n=0.6. Effect of the asymmetric heating is apparent in the region where the fluid temperature is minimum, and position of the minimum changes according to n. Figure 3 shows the maps of the dimensionless volumetric rate of entropy generation J for n=o2 and 0.6

7 386 Y. Demirel, H.H. AI-Ali and B.A. Abu-AI-Saud Vol. 24, No J O. <~> ~ 0.~'2"---. I/ y0"6 0.8~ ~ 0. 8 ~ 0.6 J O. y0.6 FIG. 3 The distribution of volumetric rate of entropy generation J for the rectangular packed duct packing: (a) n=02 " (b) n=0.6 The trends in the temperature and entropy distributions are similar. The values of J also show a minimum where temperature profiles indicate a minimum, and J is high near the heated walls. In such regions the values of qb are minimum and entropy generation due to heat transfer dominates as seen in Fig. 4.

8 Vol. 24, No. 3 ENTROPY GENERATION IN A HEATED PACKED DUCT 387 o ~ (a) 20.40"~ ~ (b) yo. 60~ FIG 4 Distributions of ratio of entropy generations due to finite pressure and temperature,b: (a) n=0.2 ; (b) n=0,6 However, according to the values of n, qb produces maximum where the temperature profile reaches minimum, hence the heat transfer is minimum. As the total heat input increase, the peak value ofdp decreases, as seen in Fig. 4b. In Fig. 5 the effects of Re on J' are shown for n=02 and 0.6. The cross-sectional entropy generation decreases as Re increases at low values of X

9 388 Y. Demirel, H.H. AI-Ali and B.A. Abu-AI-Saud Vol. 24, No. 3 j, 0.2, 0.21 (a) Re FIG. 5 Effect of Re on cross-sectional entropy generation, J' for the duct (a) n=0.2 (b) n=0.6. Conclusions An expression for the volumetric rate of entropy generation of convection heat transfer in an asymmetrically heated rectangular packed duct has been derived and displayed graphically for a fully developed velocity and temperature profiles. The effect of asymmetric thermal conditions on the entropy generation have been shown. The maps of the volumetric entropy generation reveal the thermodynamic behavior of the transfer system and leads to a better understanding and design of such systems for a specified heat transfer task.

10 Vol. 24, No. 3 ENTROPY GENERATION IN A HEATED PACKED DUCT 389 Acknowledgments The authors wish to acknowledge King Fahd University of Petroleum & Minerals for funding the research project No. CHE/HEATFLUX/173 that resulted in generating this research. A Nomenclature parameter given by equation (2), dimensionless Cp specific heat at constant temperature, J kg "1 K -1 D Dp~ D e equivalent diameter of duct, m I3.57 (WH)21 '3 L W+HI h average heat transfer coefficient, W m -2 K "1 H depth of duct, m S"' kei z dimensionless J' cross-sectional entropy generation, dimensionles kf thermal conductivity &fluid, W m -l K -l k e effective thermal conductivity of fluid, W m -1 K -I L flow path length, m n Q2/Q1 Nu Nusselt number, Nu=hDe/kf Q heat flux rate, W m "2 X x/h (;Dp Rep Reynolds number, Re -- ju Re Reynolds number, Re -- GI) ~ (1 - s)p S*tp St T U W X volumetric rate of entropy generation, W m -3 K-I Stanton number, St - temperature, K velocity, m s -1 width of duct, m direction of fluid flow h pll~, C p

11 390 Y. Demirel, H.H. AI-AIi and B.A. Abu-AI-Saud Vol. 24, No. 3 y direction normal to the flow direction o~ effective thermal diffusivity, m s 2 e void fraction la Newtonian fluid viscosity, kg m "l s -1 p density, kg m -3 d~ ratio of entropy generation by friction to that of heat transfer "t entrance temperature number, ~ = Q l/(hto), dimensionles References 1. A.P. Colburn, Heat transfer and pressure drop in empty, baffled, and packed tubes, Ind. Eng. Chem. 23, (1931). 2. Y. Demirel, Experimental investigation of heat transfer in a packed duct with unequal wall temperatures, Exp. Thermal & Fluid Science, 2, (1989). 3. G. M Chrysler and R E. Simons, An experimental investigation of the forced convection heat transfer characteristics of Fluorocarbon liquid flowing through a packed-bed for immersion cooling of microelectronic heat sources, ASME Symp. HTD 131, (1990). 4. T.H. Hwang, Y Cai and P. Cheng, An experimental study of forced convection in a packed channel with asymmetric heating, Int. J. Heat Mass Transfer, 35, (1992). 5. S.X. Gao and J. P Hartnett, Analytical Nusselt number predictions for slug flow in rectangular duct, Int. Comm. Heat Mass Transfer, 20, (1993). 6. Y. Demirel and H. H. AI-AIi, Performance Analysis of Convective Heat Transfer in a Packed Duct with Asymmetrical Wall Temperatures, Int. J. Heat Mass Transfer, in press. 7. A. Bejan, Advancedengmeering lhermodynamics, p John Wiley, New York (1988). 8. D. Kulkarni and L. K Doraiswamy, Estimation of effective transport properties in packed bed reactors, Catal. Rev. Sci. Eng., 22, (1980). 9 W.M. Kays and M. E Crawford, Convective Heat andmass Transfer, 2nd Ed p McGraw-Hill, New York (1980). Received August l 7, 1996

Thermodynamic analysis of convective heat transfer in an annular packed bed

University of Nebraska - Lincoln From the SelectedWorks of YASAR DEMIREL 000 Thermodynamic analysis of convective heat transfer in an annular packed bed YASAR DEMIREL R Kahraman Available at: https://works.bepress.com/yasar_demirel/38/